


“HDD “Uo}}D055 
“A'N ‘asnon Ag 
OU] "SOUd GUYOTAWS ( 
Aq pesnyooynuow 
~~! 


Y30NId LIIHdWyd 
auvogss3ud 








The person charging this material is re- 
sponsible for its return on or before the 


Latest Date stamped below. 


Theft, mutilation, and underlining of books 
are reasons for disciplinary action and may 
result in dismissal from the University. 


University of Illinois Library 





DEC 30 1970 


a] 


po 


L161— 0-1096 








ee eae: IN 1911 











+ if . f BY 


J. HADAMARD 


MEMBER or THH INSTITUTH, PROFHSSOR IN THE COLLEGE DBE FRANCE AND IN THH ECOLE POLYTECHNIQUE, 
LECTURER IN MATHEMATICS AND MATHEMATICAL PHYSICS IN COLUMBIA UNIVERSITY FOR 1911 








NEW YORK 
COLUMBIA UNIVERSITY PRESS 
1915 








COLUMBIA UNIVERSITY IN THE CITY OF NEW YORK 


PUBLICATION NUMBER FIVE 
OF THE ERNEST KEMPTON ADAMS FUND FOR PHYSICAL RESEARCH 


ESTABLISHED DECEMBER 17 tx, 1904 


FOUR LECTURES 
ON MATHEMATICS 


DELIVERED AT COLUMBIA UNIVERSITY 
INS {or 


BY 


J. HADAMARD 


MEMBER. OF THE INSTITUTE, PROFESSOR IN THE COLLEGE DB FRANCE AND IN THE ECOLE POLYTECHNIQUE, 


LECTURER IN MATHEMATICS AND MATHEMATICAL PHYSICS IN COLUMBIA UNIVERSITY FOR 1911 





NEW YORK 
COLUMBIA UNIVERSITY PRESS 
1915 


* 


Coryricut 1915 BY Conumpra University Press 


at 

_PRESS OF _ 

THE NEW ERA PRINTING COMPANY 
LANCASTER, PA. 


1915 





~ 


IE 


On the seventeenth day of December, nineteen hundred and four, Edward Dean 
Adams, of New York, established in Columbia University ‘‘The Ernest Kempton 
Adams Fund for Physical Research”? as a memorial to his son, Ernest Kempton 
Adams, who received the degrees of Electrical Engineering in 1897 and Master of 
Arts in 1898, and who devoted his life to scientific research. The income of this 
fund is, by the terms of the deed of gift, to be devoted to the maintenance of a 
research fellowship and to the publication and distribution of the results of scien- 
tific research on the part of the fellow. A generousinterpretation of the terms of the 
deed on the part of Mr. Adams and of the Trustees of the University has made it 
possible to issue these lectures as a publication of the Ernest Kempton Adams Fund. 








Publications of the 
Ernest Kempton Adams Fund for Physical Research 


Number One. Fieldsof Force. By VitHetmM FrimMAan Kornn Bsyerxnss, Professor of Physics 
in the University of Stockholm. A course of lectures delivered at Columbia Univer- 
sity, 1905-6. 


Hydrodynamic fields. Electromagnetic fields. Analogies between the two. Supplementary lecture on 
application of hydrodynamics to meteorology. 160 pp. 


Number Two. The Theory of Electrons and its Application to the Phenomena of Light and 
Radiant Heat. By H. A. Lormntz, Professor of Physics in the University of Leyden. 
A course of lectures delivered at Columbia University, 1906-7. With added notes. 
332 pp. Edition exhausted. Published in another edition by Teubner. 

Number Three. Eight Lectures on Theoretical Physics. By Max Puanck, Professor of 
Theoretical Physics in the University of Berlin. A course of lectures delivered at 
Columbia University in 1909, translated by A. P. Wrus, Professor of Mathematical 
Physics in Columbia University. 


Introduction: Reversibility and irreversibility. Thermodynamic equilibrium in dilute solutions. 
Atomistic theory of matter. Equation of state of a monatomic gas. Radiation, electrodynamic theory. 
Statisticaltheory. Principle ofleast work. Principle of relativity. 130 pp. 


Number Four. Graphical Methods. By C. Runas, Professor of Applied Mathematics in the 
University of Géttingen. A course of lectures delivered at Columbia University, 
1909-10. 


Graphical calculation. The graphical representation of functions of one or more independent variables. 
The graphical methods of the differential and integral ca'culus. 148 pp. 


Number Five. Four Lectures on Mathematics. By J. Hapamarp, Member of the Institute, 
Professor in the Collége de France and in the Ecole Polytechnique. A course of lectures 
delivered at Columbia University in 1911. 


Linear partial differential equations and boundary conditions. Contemporary researches in differen- 
tial and integral equations. Analysis situs. Elementary solutions of partial differential equations 
and Green’s functions. 53 pp. 


Number Six. Researches in Physical Optics, Part I, with especial reference to the radiation 
ofelectrons. By R. W. Woop, Adams Research Fellow, 1913, Professor of Experimental 
Physics in the Johns Hopkins University. 134pp. With 10plates. Edition exhausted. 

Number Seven. Neuere Probleme der theoretischen Physik. By W. Wren, Professor of 
Physics in the University of Wiirzburg. A course of six lectures delivered at Columbia 
University in 1913. 


Introduction: Derivation of the radiation equation. Specific heat theory of Debye. Newer radiation 
theory of Planck. Theory of electric conduction in metals, electron theory for metals. The Einstein 
fluctuations. Theory of Réntgen rays. Method of determining wavelength. Photo-electric effect and 
emission of light by canal ray particles. 76 pp. 


These publications are distributed under the Adams Fund to many libraries 
and to a limited number of individuals, but may also be bought at cost from the 
Columbia University Press. 


Digitized by the Internet Archive 
in 2021 with funding from 
University of Illinois Urbana-Champaign 


httos://archive.org/details/fourlecturesonma0Ohada_0 


PREFACE 


The “Saturday Morning Lectures ”’ delivered by Pro- 
fessor Hadamard at Columbia University in the fall of 
1911, on subjects that extend into both mathematics and 
physics, were taken down by Dr. A. N. Goldsmith of the 
College of the City of New York, and after revision by the 
author in 1914 are now published for the benefit of a wider 
audience. The author has requested that his thanks be ex- 
pressed in this place to Dr. Goldsmith for writing out and 
revising the lectures, and to Professor Kasner of Columbia 


for reading the proofs. 


iii 













CONTENTS 


Lecture I. The Definition of Solutions of Linear Partial 
Differential Equations by Boundary Con- 
ditions. 


Lecture II. Contemporary Researches in Differential 
Equations, Integral Equations, and In- 
tegro-Differential Equations. 


Lecture III. Analysis Situs in Connection with Corres- 
pondences and Differential Equations. 


Lecture IV. Elementary Solutions of Partial Differential 
Equations and Green’s Functions. 


|= i ie ie , pA £* -» 





LECTURE I 


THe DETERMINATION oF SoLuTIONS oF LINEAR PARTIAL D1r- 
FERENTIAL EQUATIONS BY BoUNDARY CONDITIONS 


In this lecture we shall limit ourselves to the consideration of 
linear partial differential equations of the second order. 

It is natural that general solutions of these equations were 
first sought, but such solutions have proven to be capable of 
successful employment only in the case of ordinary differential 
equations. In the case of partial differential equations employed 
in connection with physical problems, their use must be given 
up in most circumstances, for two reasons: first, it is in gen- 
eral impossible to get the general solution or general integral; 
and second, it is in general of no use even when it is obtained. 

Our problem is to get a function which satisfies not only the 
differential equation but also other conditions as well; and for 
this the knowledge of the general integral may be and is very 
often quite insufficient. For instance, in spite of the fact that 
we have the general solution of Laplace’s equation, this does 
not enable us to solve, without further and rather complicated 
calculations, ordinary problems depending on that equation 
such as that of electric distribution. 

Each partial differential equation gives rise, therefore, not to 
one general problem, consisting in the investigation of all solu- 
tions altogether, but to a number of definite problems, each of 
them consisting in the research of one peculiar solution, defined, 
not by the differential equation alone, but by the system of that 
equation and some accessory data. ; 

The question before us now is how these data may be chosen 
in order that the problem shall be “correctly set.” But what 
do we mean by “correctly set”? Here we have to proceed by 
analogy.. 


2 FIRST LECTURE 


In ordinary algebra, this term would be applied to problems 
in which the number of the conditions is equal to that of the 
unknowns. To those our present problems must be analogous. 
In general, correctly set problems in ordinary algebra are char- 
acterized by the fact of having solutions, and in a finite number. 
(We can even characterize them as having a unique solution 
if the problem is linear, which case corresponds to that of our 
present study.) Nevertheless, a difficulty arises on account of 
exceptional cases. 

Let us consider a system of linear algebraic equations: 

ayy + »+ dnt, = bh 
(1) we Gd ee er 
the number n of these equations being precisely equal to the 
number of unknowns. If the determinant formed by the co- 
efficients of these equations is not zero, the problem has only 
one solution. If the determinant is zero, the problem is in 
general impossible. At a first glance, this makes our aforesaid 
criterion ineffective, for there seems to be no difference between 
that case and that in which the number of equations is greater than 
that of the unknowns, where impossibility also generally exists. 
(Geometrically speaking, two straight lines in a plane do not 
meet if they are parallel, and in that they resemble two straight 
lines given arbitrarily in three-dimensional space.) The dif- 
ference between the two cases appears if we choose the b’s (second 
members of the equation (1) ) properly; that is, in such manner 
that the system becomes again possible. If the number of 
equations were greater than n, the solution would (in general) 
again be unique; but, if those two numbers are equal, the problem 
when ceasing to be impossible, proves to be indeterminate. 

Things occur in the same way for every problem of algebra. 
For instance, the three equations 


Fie, Y; 2) — a 
g(x, y, 2) = b 
| toe ae 


LINEAR PARTIAL DIFFERENTIAL EQUATIONS 3 


between the three unknowns 2, y, 2, constitute an impossible 
system if ¢ is not equal to a+ b, but if ¢ equals a+ 5, that 
system is in general indeterminate. 

Moreover, this fact has been both extended and made precise 
by a most beautiful theorem due to Schoenflies. 


Let 
(2) f(x,y, 2) = X, g(x, 4,2) = Y, Rare ys) = Z 


be the equations of a space-transformation, the functions f, g, h 
being continuous. Let us suppose that within a given sphere 
(27+ y? + 2 = 1, for instance), two points (2, y, 2) cannot give 
the same single point (X, Y, Z): in other words, that f(z, y, 2) 
ae fe’, y’, 2), g(x, y, 2) = g(x’, y', 2); h(x, y, 2) = h(’, y’, 2’) 
cannot be verified simultaneously within that sphere unless 
x=2',y=y',2=2'. Let S denote the surface corresponding 
to the surface s of the sphere; that is, the surface described by 
the point (XY, Y, Z) when (x, y, 2) describes 5. If in equation (2) 
we consider now X, Y, Z as given and a, y, z as unknown, our 
hypothesis obviously means that those equations cannot admit 
of more than one solution within s. Now Schoenflies’ theorem 
says that those equations will admit of a solution for any (X, Y, Z) 
that may be chosen within S. Of course the theorem holds 
for spaces of any number of dimensions. It is obvious that this 
theorem illustrates most clearly the aforesaid relation between 
the fact of the solution being wnique and the fact that that 
solution necessarily exists.1 

As said above, the theorem is in the first place remarkable for 
its great generality, as it implies concerning the functions f, g, h 
no other hypothesis but that of continuity. But its significance 
is in reality much more extensive and covers also the functional 
field. I consider that its generalizations to that field cannot 

1 We must note nevertheless, that in it the unique solution is opposed not 
only to solutions in infinite number (as above), but also to any more than 
one. For instance, the fact that 2? = X may have no solution in 2, is, from 


the point of view of Schoenflies’ theorem, in relation with the fact that for 
other values of X, it may have two solutions. 


4 FIRST LECTURE 


fail to appear in great number as a consequence of future dis- 
coveries. This remarkable importance will be my excuse for 
digressing, although the theorem in question is only indirectly 
related to our main subject. The general fact which it emphasizes 
and which we stated in the beginning, finds several applications 
in the questions reviewed in this lecture. It may be taken as a 
criterion whether a given linear problem is to be considered as 
analogous to the algebraic problems in which the number of 
equations is equal to the number of unknown. This will be the 
case always when the problem is possible and determinate and 
sometimes even when it is impossible, if it cannot cease (by 
further particularization of the data) to be impossible otherwise 
than by becoming indeterminate. 

Let us return to partial differential equations. Cauchy 
was the first to determine one solution of a differential equa- 
tion from initial conditions. For an ordinary equation such as 
f(x, y, dy/dx, Py/dx?) = 0, we are given the values of y and 
dy/dx for a particular value of x. Cauchy extended that result 
to partial differential equations. 

Let F(u, 2, y, 2, 0u/dx, du/dy, du/dz, 0’u/dx?, ---) = Obeagiven 
equation of the second order and let it be granted that we can 
solve it.with respect to 0?u/dx”. Thus we obtain (02u/d22) + Fy 
= 0 where F; is a function of all of the above quantities, except 
0’u/da?. Then Cauchy’s problem arises by giving the values 


0 
(3) w= 0,2), 5. = V2) 


of w and du/dx for x= 0. (These data must be replaced by 
analogous data if, instead of the plane x = 0, we introduce 
another surface.) Indeed, under the above hypothesis concerning 
the possibility of solving the equation with respect to 6?u/da?, 
and on the supposition that the functions Fi, ¢ and w are holo- 
morphic, Cauchy, and after him, Sophie Kowalevska, showed 
that in this case there is indeed one and only one solution. 
This solution can be expanded by Taylor’s series in the form 
U = Up + xu, + a?u2 + +++ where w, mw, --- can be calculated. 


LINEAR PARTIAL DIFFERENTIAL EQUATIONS 5 


The above theorems are true for most equations arising in 
connection with physical problems, for example 


(E) VU = OE : 

But in general these theorems may be false. This we shall 
realize if we consider Dirichlet’s problem: to determine the 
solution of Laplace’s equation 

071 4 
(¢) Yun Sat gat gaa 
for points within a given volume when given its values at every 
point of the boundary surface S of that volume. 

It is a known fact that this problem is a correctly set one: it 
has one, and only one, solution. Therefore, this cannot be the 
case with Cauchy’s problem, in which both wu and one of its 
derivatives are given at every point of S. If the first of these 
data is by itself (in conjunction with the differential equation) 
sufficient to determine the unknown function, we have no right 
to introduce any other supplementary condition. How is it 
therefore that, by the demonstration of Sophie Kowalevska, the 
same problem with both data proves to be possible? 

Two discrepancies appear between the sense of the question 
in one case and in the other: (a) In the theorem of Sophie 
Kowalevska, wu has only to exist in the immediate neighborhood 
of the initial surface S. In Dirichlet’s problem, it has to exist 
and to be well determined in the whole volume limited by S. 
We therefore require more in the latter case than in the former, 
and it might be thought that this is sufficient to resolve the 
apparent contradiction met with above. 

In fact, however, this is not the case and we must also take 
account of the second discrepancy. (b) The data, in the case of 
the Cauchy-Kowalevska demonstration, are, as we said, sup- 
posed to be analytic: the functions ¢, y (second members of 
(3)) considered as functions of y, 2, are taken as given by con- 
vergent Taylor’s expansions in the neighborhood of every point 


6 FIRST LECTURE 


of the plane x = 0 in the region where the question is to be solved, 
Nothing of the kind is supposed in the study of Dirichlet’s 
problem. Not even the existence of the first derivatives of w, 
corresponding to displacements on S, is postulated, and in some 
researches, certain discontinuities of these values are admitted. 
Both these circumstances play their réle in the explanation of 
the difference between the two results discussed above. 

That (a) is one reason for that difference is evident, for of 
course, if a function is required to be harmonic (i. e. to admit 
everywhere derivatives and to verify Laplace’s equation) within 
a sphere, its values and those of its normal derivative, may not 
together be chosen arbitrarily on the surface even if analytic. 

To show that (a) is not sufficient for the required explanation, 
let us take the geometric terms of the problem in the same way 
as Cauchy. We therefore suppose that, w being defined by 
Laplace’s equation, the accessory data given to determine it 
are the values of uw, and du/dx on the plane x = 0, or, more 
exactly, on a certain portion Q of that plane; w will also not be 
required, now, to exist in the whole space; its domain of existence 
may be limited, for instance, to a certain distance, however small, 
from our plane x = 0 (in the environs of Q) provided that 
distance be finite and not infinitesimal. 

Now under these conditions, in general such a function u 
does not exist, if the data are not analytic and are chosen arbi- 
trarily. One sees then a fact which never appeared as long as 
ordinary differential equations were alone concerned, namely, 
that the results are utterly different according as the analytic 
character of the data is postulated or not. 


3. 


Of these two opposite results which is to be considered as 
giving us a more correct and adequate idea of the nature of 
things? I do not say as the true one, for of course each one is so 
under proper specifications. 

Some mathematicians still incline to prefer the old point 


LINEAR PARTIAL DIFFERENTIAL EQUATIONS a 


of view of Cauchy, one of their reasons being that, as known 
since Weierstrass, any function, analytic or not, can be replaced 
with any given approximation by an analytic one, (more pre- 
cisely by a polynomial). Therefore the fact that a function 
belongs to one or the other of those two categories seems to them 
to be immaterial. I cannot agree with this point of view. 
That the thing is no¢ immaterial, seems to me to follow directly 
from what we have just stated. And it cannot fail to be put in 
evidence if we think not only of the mere existence of the solu- 
tion, but of its properties and the means of calculating it. If 
Cauchy’s problem, for equation (e), ceases to be possible, as a 
rule, when the functions designated by ¢, W are not analytic, 
then every expression for the solution must depend essentially 
on that analyticity and especially upon the radii of convergence 
of the developments of ¢, Y. In other words, let us imagine 
that the functions ¢, y be replaced by other functions g, yu, 
the differences ¢g; — 9, ¥%1 — W being very small for every 
system of real values of y, « within 2 (and perhaps also the 
differences of some derivatives being small). However slight 
the alteration may be it rigorously follows from the afore- 
said theorem of Weierstrass, that the radii of convergence of 
the developments in power series (if existing at all) may and 
will be, in general, completely changed; so the calculations lead- 
ing to the solution will necessarily be changed also. 

If that solution itself should undergo but a slight change, this 
would at once show us that these methods of calculation ought 
to be of quite an artificial nature, masking completely the quali- 
tative properties of the required result.1. But in fact, it is clear 
that matters are not as just assumed above. The alteration 
u, — u produced on the values of w by our slight modification 

1 The solution by development in Taylor’s series is, in general, for problems 
of that kind, the only one which can be given. I know but one exception, 
which is Schwarz’s method for minimal surfaces, when a curve of the surface 
and the corresponding succession of tangent planes are given. This method 


rests on the favorable and exceptional circumstance that complex variables 
can be employed for the study of real points of such a surface. 


8 FIRST LECTURE 


of y, W will be generally important and often complete, as is 
evident! by the fact that wu will cease completely to exist when 
¢, W become non-analytical. This proves, first of all, that the 
application of Weierstrass’ theorem in that case is illegitimate, 
since it gives an approximation for the data but nothing of the 
kind for the unknown. 

Then we see also that such a problem and calculation, the 
results of which are utterly changed by an infinitesimal error in 
starting, can have no meaning in their applications. 

This leads to my second and chief reason for considering 
only the results which correspond to non-analytic data, namely, 
the remarkable accordance between them and the results to 
which physical applications bring us. 

This accordance is the more interesting from the fact of its 
results being unexpected. Our former point of view—i. e. that 
of the Cauchy-Kowalevska theorem—evidently constitutes a 
complete analogy to the case of ordinary differential equations. 
But from our latter point of view—which is also the point of 
view in problems set by physical applications—every analogy 
seems to be upset. The results often seem almost inco- 
herent, they will give opposite conclusions in apparently similar 
questions. 

A first instance of this was given above. We know that 
Cauchy’s problem is now impossible for Laplace’s equation 
() au = Fat sat gan 0s 
but, on the contrary, in the equation of spherical waves 


Ou Ou. dui du 
(E) BE apie ar ae op 


11f wu; — wu should be uniformly very small at the same time as yg: — ¢, 
vi — w, it follows from the well-known convergence theorem of Cauchy that, 
letting the analytic functions gi, yi, converge towards certain (non-analytic) 
limiting functions ¢, y, the corresponding solution wu ought to converge 
uniformly towards a certain limit uw, which would be solution of the problem 
with the data ¢, y. 





LINEAR PARTIAL DIFFERENTIAL EQUATIONS 9 


or of the cylindrical waves 


(E”) ee 0 

Otmea Ome Ol: : 
we may assign arbitrarily the values (whether analytical or not) 
of w and 6u/é¢ for = 0, and Cauchy’s problem set in that way 
has a solution (which is unique). In this latter case it is like 
a problem in algebra in which the number of equations is equal 
to the number of unknowns; in the former, like a problem in 
which the number of equations is superior! to the number of 
unknowns. 

It never could have been imagined a priori that such a difference 
could depend on the mere changing of sign of a coefficient in 
the equation. But it is entirely conformable to the physical 
meaning of the equations. Equation (E’), for instance governs 
the small motions of a homogeneous and isotropic medium, like a 
homogeneous gas; and the corresponding Cauchy’s problem, 
enunciated above, represents the definition of the motion by 
giving the state of positions and speeds at the origin of times. 
On the contrary, equation (e), which also governs many physical 
phenomena, never leads to problems of that kind but exclusively 
to problems of the Dirichlet type. The analytical criterion by 
which those two kinds of partial differential equations are to be 
distinguished, is known: it is given by what are called the 
characteristics of an equation. The characteristics of an equation 
correspond analytically with what the physicist calls the waves 
compatible with this equation, and are calculated in the following 
way. Let a wave be represented by the equation P(a, y, 2, 2) 

1 We could be tempted to apply in that case the remark made in the be- 
ginning (p. 4) concerning such impossible problems, which, notwithstanding 
that circumstance, must be considered as resembling “correctly set’’ ones. 
This, however, is not really applicable; for we have seen that the category 
alluded to is recognized by the fact that the problem may, under more special 
circumstances, become indeterminate. Now, this can never be the case in 
the present question: it follows from a theorem of Holmgren (“ Archiv fiir 


Mathematik”’) that the solution of Cauchy’s problem, if existent, is in every 
possible case unique. 


10 FIRST LECTURE 


= 0. Inthe given equation, for instance, if A2v — 1/a?- 0?u/dt2 = 0 
and A’u be replaced by (@P/dx)2+ (0P/dy)? + (OP/dz)? and 
— (1/a*)(@u/de) by — (1/a2) (0P/dt)* the condition thus obtained 


1S 
oP \2 OP \2 OP\2" W1; L.0P \? 
es HG) alee) -a() =° 


(which is a partial differential equation of the first order). 
It must be verified by the function P. When ‘this holds, 
P(x, y, 2,t) = Ois said to be a characteristic of the given equation. 
For equation (E), such characteristics exist (that is, are real); 
this case is called the hyperbolic one. 
Laplace’s equation, A2u~ = 0, on making the above substitu- 
tion, leads to the equation 


die Olas dns? 
(2) + (Ys ey 


which has no real solution. Therefore, in this case there are no 
waves and we have the so-called elliptic case. Cauchy’s problem 
can be set for a hyperbolic equation, but not for an elliptic one. 
Does this mean that for a hyperbolic equation Cauchy’s problem 
will always arise? No, the matter is not quite so simple. For 
instance, in equation (E) or (E’), we could not choose arbi- 
trarily w and du/dy for « = 0; this would lead us again to an 
impossible problem (in the non-analytic case, of course). 

The physical explanation of this lies in the fact that there are, 
besides the partial differential equation, two kinds of conditions 
determining the course of a phenomenon, viz., the initial and the 
boundary conditions. The former are of the type of Cauchy 
and they alone intervene in Cauchy’s problem quoted above 
for the equation of sound. 

But the boundary conditions are always of the type of Dirich- 
let. They are the only ones which can occur in an elliptic 
equation, but even in a hyperbolic one they generally present 


1 An intermediate case exists A2u — k(du/at) = 0. This is semi-definite 
and is termed the parabolic one (example: the equation of heat). 


LINEAR PARTIAL DIFFERENTIAL EQUATIONS if 


themselves together with initial ones. This gives place to so- 
called mixed problems where the two kinds of data (belonging 
respectively to the Cauchy and to the Dirichlet type) intervene 
simultaneously for the determination of the unknown. 

In equation (Z), ¢ = 0 represents the origin of time and can 
give place to initial conditions, having the form, of Cauchy. 
But no such conditions can correspond to 2 = 0, which represents 
a geometric boundary. 

More or less complicated cases can arise for various disposi- 
tions of the configurations, giving place to other paradoxical 
and apparently contradictory results, which can however all be 
explained in the same way. Moreover, there are other types 
of linear partial differential equations,’ which do not govern any 
physical phenomena. The determination of solutions has been 
studied? in the analytic case but no sort of determination of 
that kind for non-analytic data has been discovered hitherto. 

We see that from this non-analytic point of view the accord- 
ance between mathematical results and the suggestions of 
physics holds perfectly. This accordance must not surprise us, 
for, as we saw above, it corresponds to the faet that a problem 
which is possible only with analytic data can have no physical 
meaning. But it remains worth all our attention. No other 
example better illustrates Poincaré’s views*® on the help which 
physics brings to analysis as expressed by him in such statements 
as the following: “It is physics which gives us many important 
problems, which we would not have thought of without it,” 
and “It is by the aid of physics that we can foresee the solutions.” 


1 The so-called non-normal hyperbolic equations, such as 


Pu Ou Pu CEN 

aaa ee wise eats Sig ee 
2 By Hamel (Inaugural Dissertation, Géttingen) and Coulon (thesis, Paris) 
3 Lectures delivered at the first International Mathematical Congress, 


Zurich, 1897; reproduced in ‘‘ La Valeur de la Sciences.” 


LIBRARY 
UNIVERSITY OF ILLINOIS 


LECTURE II 


CONTEMPORARY RESEARCHES IN DIFFERENTIAL EQUATIONS, 
INTEGRAL EQuaTIons, AND INTEGRO-DIFFERENTIAL 
EQuaATIONS 


1. Partial Differential Equations and Integral Equations 


T reminded you at the end of the last lecture what indispensable 
help the physicist renders to the mathematician in furnishing 
him with problems. But that help is not always free from 
inconveniences, and the task of the mathematician is often a 
thankless one. Two cases generally occur: it may happen that 
the physical problem is easily soluble by a mere “rule of three” 
method, but if not, it is so extremely difficult that the mathe- 
matician despairs of solving it at all; and he will strive after 
that solution for two centuries and, when he obtains it, our 
interest in the particular physical problem may have been lost. 
Such seems to be the case with some problems concerning partial 
differential equations. Just after the discovery of infinitesimal 
calculus, physicists began by needing only very simple methods 
of integration, the problems in general reducing to elementary 
differential equations. But when higher partial differential 
equations were introduced, the corresponding problems almost 
immediately proved to be far above the level of those which 
contemporary mathematics could treat. 

Indeed, those problems (such as Dirichlet’s) exercised the 
sagacity of geometricians and were the object of a great deal of 
important and well-known work through the whole of the 
nineteenth century. The very variety of ingenious methods 
applied showed that the question did not cease to preserve its 
rather mysterious character. Only in the last years of the 


century were we able to treat it with some clearness and under- 
12 


CONTEMPORARY RESEARCHES IN EQUATIONS is: 


stand its true nature. This clearness seemed to come too late, 
for at that time, physics began its present evolution in which it 
seems to disregard partial differential equations and to come 
back to ordinary differential equations, but of course in prob- 
lems profoundly different from the simple cases which were 
familiar to Bernouilli or Evler. 

Happily, for it would have been a humiliating thing to work so 
uselessly, this disregard was only in appearance, and the ancient 
problems have not lost their importance by the fact that other 
ones have been superposed on and not substituted for them. 
In fact, the solution now obtained for Dirichlet’s problem has 
proved useful in several recent researches of physics. 

Let us therefore inquire by what device this new view of 
Dirichlet’s problem and similar problems was obtained. Its 
peculiar and most remarkable feature consists in the fact that 
the partial differential equation is put aside and replaced by a 
new sort of equation, namely, the integral equation. This new 
method makes the matter as clear as it was formerly obscure. 

In many circumstances in modern analysis, contrary to the 
usual point of view, the operation of integration proves a much 
simpler one than the operation of derivation. An example of 
this is given by integral equations where the unknown function 
is written under such signs of integration and not of differentia- 
tion. The type of equation which is thus obtained is much 
easier to treat than the partial differential equation. 

The type of integral equations corresponding to the plane 
Dirichlet problem is 


(1) OLR f b(y) K(x, v)dy = fle) 


where ¢ is the unknown function of x in the interval (4, B), f 
and K are known functions, and ) is a known parameter. The 
equations of the elliptic type in many-dimensional space give 
similar integral equations, containing however multiple integrals 
and several independent variables. Before the introduction of 


14 SECOND LECTURE 


equations of the above type, each step in the study of elliptic 
partial differential equations seemed to bring with it new diffi- 
culties; not only did the various methods imagined for Dirichlet’s 
problem not cast more than a partial light on the question, 
but the principles of most of them were peculiar to that special 
problem: they seemed to disappear if Laplace’s equation was 
replaced by any other equation of the same type, or even (except 
for Neumann’s method, which, as we shall soon see, is directly 
related to integral equations) if for the same Laplace’s equa- 
tion Dirichlet’s problem was replaced by any analogous one 
such as presented by hydrodynamics or theory of heat. Each 
of them, besides, was rather a proof of existence than a method 
of calculation. 

Then they seemed again quite insufficient for another series 
of questions which mathematical physics had to solve, viz., the 
study of harmonics. The existence of those harmonics (such as 
the different kinds of resonance of a room filled with air) was 
physically evident, but for the mathematician it offers an im- 
mense difficulty. Schwarz, Picard and Poincaré gave a first 
solution which was rather complicated as each harmonic requires 
for its definition a new infinite process of calculation after the 
preceding one has been determined. Nevertheless it has demon- 
strated rigorously the chief properties of the quantities in ques- 
tion (namely, certain special values of the parameter in equation 
(1)), i. e. that they exist and form a discrete infinity, only a finite 
number of them lying within any finite interval. 

But at the same time a discovery even more important, in a 
certain sense, was made by Poincaré, namely the near relation 
between that question of harmonics and the method which had 
been indicated by Neumann for Dirichlet’s problem. This 
discoverv of Poincaré paved the way for Fredholm’s work. The 
latter treats every one of the aforesaid questions, and any 
which can be assimilated to them, by one and the same method, 
which consists in tee reduction to an equation such as (1). 
This gives all the required results at once and for all the possible 
types of such problems, 


CONTEMPORARY RESEARCHES IN EQUATIONS 15 


In all this, the mathematician seems to play again the 
unfortunate role we alluded to in the beginning; for those 
results are nothing but the mathematical demonstration of facts 
each of which was familiar to every physicist long before the 
beginning of all those researches. But of course their interest 
is not in fact limited in demonstration; they can and do serve 
as starting points for the discovery of new facts. They are 
useful as giving the proper method of calculation. Previously, 
in the calculation of the resonance of a room filled with air, 
the shape of the resonator had to be quite simple, which require- 
ment is not a necessary one for the case where integral equations 
are employed. We need only make the elementary calculation 
of the function K and apply to the function so calculated the 
general method of resolution of integral equations. 

There are two chief methods for the solution of the equa- 
tions. It is not always easy to get numerical results. 

Liouville and Neumann (in solving Dirichlet’s problem) 
really worked out a method of solving integral equations. A 
second method is due to Fredholm. The first method leads to 
series which may converge slowly but they are easy to calculate. 
The method of Fredholm gives a quotient of two series (entire 
functions of \) the terms of which have to be calculated inde- 
pendently, while in the first method each is obtained from the 
one immediately preceding it. While we must add that Erhard 
Schmidt has shown how the first method can be made to supply 
a more rapidly convergent series, Fredholm’s method is of 
greater value to physics because of the theoretical point of view. 
It gives easily (what was impossible before its appearance) not 
only the existence of harmonics, but their properties. For 
instance, older methods could not have succeeded, at least not 
without great difficulties and a large amount of calculation, in 
obtaining the order of magnitude of the successive upper har- 
monics (i. e. the corresponding great values of \). They would 
probably have been quite unable to predict the order of magni- 
tude, as is done in the recent works of Hermann Wey], so as to 


16 SECOND LECTURE 


show its relation the volume of the room to which they 
correspond. But it has even proved of great importance for 
physics to know mathematically, and not only empirically, that 
the harmonics corresponding to equations of the form (1) are a 
discrete infinity. For in the case of the spectral frequencies we 
get series which tend to accumulate towards definite positions. 
Since Fredholm’s theory we can assert that such series are not 
compatible with the form of integral equation given at the 
beginning of this lecture. 

Fredholm himself investigated new forms (as also did Walther 
Ritz). The introduction of the integral equation has made even 
the above problem accessible. The older method would not have 
been able to decide whether the distribution in question was pos- 
sible or not. The hypothesis proposed by Fredholm leads to an 
integral equation such as 


1 b 
o@) -— pf oWKG, vdy = $@) (2) 


Here the frequencies will accumulate in the neighborhood of 
A= VK. 

I must immediately add that, as Ritz showed, Fredholm’s type 
is not sufficient to give a correct explanation of the phenomena. 
But this does not change the essential fact that by the aid of the 
new method we are immediately able to decide what the asymp- 
totic distribution of harmonics can or cannot be, so that com- 
parison with observation becomes possible; and this we owe 
entirely to Fredholm’s method. 


2. Coming Back to Ordinary Differential Equations 


As we said in the beginning, the subject of partial differential 
equations which was the main and almost the only occupation 
of mathematical physics, ceases nowadays to be so. As a con- 
sequence of the general admission of the discrete structure of 
matter, physical problems tend now to lead to ordinary differ- 
ential equations. These differential equations are to be studied 


CONTEMPORARY RESEARCHES IN EQUATIONS 17 


under the most difficult circumstances because we must follow 
the form of the solutions for very long periods of time, that is, 
of the independent variable ¢. One can say that such a study 
did not exist before Poincaré, and even his researches on the 
subject, I mean especially his four chief memoirs in the “ Journal 
de Mathematiques,” 1887 (On the shape of Curves Defined by 
Differential Equations), lead us, like Socrates, to begin to feel 
that we know nothing. 

We cannot, in this place, lay stress on the extraordinary com- 
plications and paradoxes which he discovered. We shall mention 
only one of them, because it helps to correct an error frequently 
committed in hydrodynamical and electrical problems, concern- 
ing the lines of force and the lines of flow. These lines are all 
defined by ordinary differential equations. The general form 
is dx/X = dy/Y = dz/Z. In a very general category of cases 
the vector X YZ has the property that 





Oe ets OX OY 2) OZ, 
div (XYD) = (F ay 3) =0 
Now, whenever such conditions existed, physicists used to say 
that the tubes of foree—or tubes of flow, or tubes of vortices— 
were closed (if they did not go to infinity or come to the 
boundaries of the domain of existence of the vector XY, Y, Z). 

They were, I think, led to say so by the examples given by 
some simple peculiar cases in which the differential equations 
could be integrated, for one could not suspect before Poin- 
caré’s work that such cases are exceptional, generally giving 
a quite inadequate and deformed view of things. In fact, the 
assertion in question is an utterly false one.’ If you allow me 
such a crude comparison, it is not true that the tube of force 
must get back home and put its key in the lock. Rather does 
it put its key above and below and on either side, and never 
succeeds in getting it in exactly. It will, it is true, nearly get 


1A demonstration is frequently given to justify it, the error of which 
consists in an incomplete enumeration of possible cases. 


18 SECOND LECTURE 


back an infinite number of times. The only consequence which 
can be correctly drawn from the equation div(X YZ) = 0 is 
that the area of the cross section of the tube cannot have changed. 
But its shape may, and generally will, have done so. If it were, 
let us say, circular in starting, it will have become elliptic when 
coming back and its ellipticity will increase at each return. 
Finally it will become a long flat strip and only a part of it will 
come back to the neighborhood of its original position. In Fig. 
1, the successive appearances of the same tube of force are shown. 





Fig. 1 


The tube of force may have been originally circular, but on its 
first recurrence or return, it may have become elliptic in cross 
section and thus it has only partly returned to its original 
position. Still more is this the case in the second recurrence of 
the tube of force, which may be assumed by this time to have 
become very flat in cross section. 

As Mr. Birkhoff kindly pointed out to me, it is interesting 
to remark that in most cases, the deformed and flattened tube 
will even pass simultaneously indefinitely near to any point of 
the considered medium. 

A rather curious fact must nevertheless be stated. Although 
the principle that the tube is closed is completely false, the 


CONTEMPORARY RESEARCHES IN EQUATIONS 19 


conclusions drawn from it by physicists are most often true. 
Why is this so? Perhaps the explanation lies in the fact that 
under that same hypothesis, div (YX, Y, Z) = 0, a line defined 
by our differential equations generally returns indefinitely near 
and an infinite number of times to its starting point. (This is 
called “‘ Stabilité a la Poisson.’’) Poincaré has shown that though 
not every line in question necessarily does this, the fact occurs 
for an infinitely greater number of cases than those in which it 
does not occur. 


3. Application to Molecular Physics 


We see by this single example how complicated and unexpected 
the shapes of curves defined by differential equations may be, 
and how far we are from understanding them when considered 
for great values of the independent variable. 

But could we be satisfied with our work if we succeeded in 
doing so? This even is doubtful. I cannot help thinking of 
a bequest left to the French Academy of Science for a prize to 
the first person who should be able to communicate with a 
planet other than Mars! The case of molecular physics reminds 
me of that rather difficult requirement. The discussion of the 
molar effects (i. e. the effects on quantities of matter accessible 
to observation) of molecular movements is a mathematical 
problem, which, logically speaking, would presuppose a rather 
advanced knowledge of curves defined by differential equations, 
and take this as a starting point, in order to discuss the questions 
of probability connected with such curves. 

That probability plays its role in the movements of almost any 
dynamical system, follows from the statements we just quoted. 
If the initial positions and the initial speeds of the moving points 
are exactly given, so will be the final positions and speeds after 
any (however long) given period of time. But if this period is 
long, and if we make a very small error in the initial conditions, 
the small error will have a much magnified effect and even cause 
a total change in the results at the end of the long period of 
time, and this is precisely Poincaré’s conception of hazard. 


20, SECOND LECTURE 


It is like a roulette game at Monte Carlo where we do not know 
all the conditions of launching the ball which induces the hazard. 
And so we know nothing more about the conditions than the 
gamblers. In other words, molecules are finally mixed just as 
cards after much shuffling. It is this fundamental hazard which 
plays the main part in Gibbs’s method. A sort of mixing func- 
tion ought to be introduced. Let us start on one of the lines of 
force. If we know exactly the point of departure A we should 
know accurately the point of arrival. If A is but approximately 
known, that point of arrival may occupy all sorts of positions; 
and indeed, in many differential problems, it may coincide 
(approximately) with any point B within the domain where the 
differential system is considered (though this is not exactly so 
for dynamical problems on account of the energy integral or 
other uniform integrals which the equations may admit). 

Therefore, the starting point being approximately A, there 
will be a certain probability that the point of arrival will be in a 
certain neighborhood of another given point B; and that prob- 
ability will be a certain function of the positions of the two 
points A, B. 

Now, logically speaking, in order to solve the question set 
for us by kinetic theories, we ought to take such a “mixing 
function,” assuming it to be known, as a base for further and 
perhaps complicated reasoning. In fact, the main present 
theories in statistical mechanics rest on certain assumptions 
concerning that function, which are very plausible. But, rigor- 
ously speaking, we are not able to consider them as theorems. 

Happily, things are greatly simplified by the fact that in-such 
mixings the aforesaid function, characteristic of the law of 
mixing, only intervenes by some of its properties and may be 
changed to a large extent without changing the final result. 
This is what Poincaré showed for the ordinary shuffling of cards 
in his “Calcul des Probabilités” (second edition). In one 
shuffling the peculiar habits of the player certainly intervene 
and so do they more or less after only a few shufflings. But 


CONTEMPORARY RESEARCHES IN EQUATIONS 21 


after many shufflings the results become totally independent of 
those habits. Poincaré also shows (though with some excep- 
tions which do not however seem to play a great practical réle), 
that such is likewise the case in the kind of mixing introduced by 
molecular theories. 

Some known facts in the history of these theories give a 
striking instance of this. Such is the work of Boltzmann and 
Gibbs in the treatment of the kinetic theory of gases and 
statistical mechanics. They both obtained the result that if 
we consider the probability of the average number of mole- 
cules in 6-dimensional space and call it P, and integrate log P 
over the whole mass, the conclusion drawn will be that the 
integral obtained is constantly increasing. Critics, and among 
them my colleague and friend Brillouin, say: “We bave not 
to congratulate ourselves on the result, because the two speak 
of quite different things and yet they agree. Gibbs does not 
mention the collision of molecules, while Boltzmann’s analysis 
is founded on the collisions of molecules. The primitive order 
of the molecules is disturbed by such collisions and a mixing is 
produced. Gibbs gets a similar mixing by the mere considera- 
tion of differential equations existing over long periods of time.” 
In both cases, if we consider systems which are “molecularly 
organized,” after a certain time the molecules will be so much 
less organized and more mixed up. 

We are surprised to find this coincidence of the results of 
Gibbs and of Boltzmann in such circumstances. We shall, how- 
ever, cease to consider it as fortuitous and perceive its true 
signification by precisely what we just remarked on the shuffling 
of cards, which makes us understand that such final results may 
and do depend on properties which are, in general, common to 
utterly various laws of mixing. 

But the difficulties met with in partial or ordinary differential 
equations are not the only ones which we had to consider at the 
present time. The mathematicians have contrived to introduce 
a new sort of equation, more difficult than the previous ones, the 
integro-differential equation. 


22 SECOND LECTURE 


4, Integro-differential Equations 

Weare now forced to consider this newform. Here the unknown 
function simultaneously appears in integrals and in differentials. 
We have at least two completely different cases of such equations 
to consider. Their difference corresponds to the two sorts of 
variables which intervene in all physical problems, the space 
variables x, y, z, and the time variable ¢t. (There may be more 
than three variables in the first group.) 

Type 1: Differentiation with respect to a, y, z; integration 
relative tot. Type 2: Differentiation with respect to ¢; integra- 
tion relative to 2, y, z. And even though this type dates only 
from 1907, we have already found cases of both kinds. 

Volterra was led to consider the first one in connection with 
“The Mechanics of Heredity.” This is the case where the 
properties of the system depend on all the previous facts of its 
existence (such as magnetic hysteresis, strains of glass, and 
permanent deformations in general). 

Volterra considers elastic hysteresis. Let 7’ be any component 
of strains; EZ the component of deformation. (There are six 7’s 
and six E’s.) Then formerly we considered 7),,= Da;,E,,. There 
are 6 equations of this type. There are 21, 36, 6 or 2 a’s depend- 
ing on the theories. If we consider heredity, we must introduce 
new terms. Suppose that at the time 0 there were xo strains; then 


t 
Tre = Dak, + i (ZaE) dr where t is the variable time. This 
0 


is an equation in which we have derivatives with respect to 2, 
y, 2, and an integral with respect to the time; and the same 
character subsists if, from those values of the 7’s, we deduce 
the equations of movement. Water waves furnish us with an 
instance of the opposite type. One knows that waves on the 
surface of water are the most common example of an undulatory 
phenomenon and that, for this reason, they are most frequently 
used to give to the beginner a first idea of what such phenomena 
are. 

But it is a general, though astonishing fact, that the most 


CONTEMPORARY RESEARCHES IN EQUATIONS 23 


simple of daily phenomena are the most difficult to understand. 
While the theory of aérial or even elastic waves is rather simple, 
at least as long as viscosity is left aside,’ and now classically 
reduced to analytical principles (related to notion of characteris- 
tics as we saw in the preceding lecture), the properties of surface 
waves in liquids are much more hidden. The few results clas- 
sically known on that subject are even of a contradictory nature. 
One of them is the differential equation given by Lagrange in 
the case of small (and constant) depth, which has served as a 
model for the dynamical theory of tides, the equation obtained 
as governing the phenomenon being in both cases a partial 
differential equation of the second order. But, for the same 
phenomenon on a liquid of indefinite depth, Cauchy gets a 
partial equation of the fourth order. The truth is that the 
problem does not lead to a differential equation at all, but to 
an integro-differential equation. For an originally plane surface 
with small displacements, where z is the vertical displacement 


at (x, y), then 
d’z 
Ta | | Za(P, QaSq 


Thus, for any determinate point P of the surface defined by its 
codrdinates, (x, y), the vertical acceleration depends on the 
values of z in every other point Q(z’, y’). Here Sg is dx’dy’ 
and ¢ is a known function of (2, y, 2’, y’). The above equation 
is of the second form of integro-differential equations. 

Volterra succeeded in the case of isotropic bodies in reducing 
the problem to the solution of a partial differential equation and 
an ordinary integral equation. But things are not so simple 
for crystalline media.” 

1 In a viscous gas, waves cannot exist, strictly speaking. They are replaced 
by quasi-waves which were first considered by Duhem, and more profoundly 
studied in an important memoir presented by Roy to the French Academy 
of Sciences. 

Since these lectures were delivered, Professor Volterra has given a com- 
prehensive view of his methods and solutions in a course of lectures at the 


University of Paris. See the issue of those lectures by J. Peres (Paris, Gauthier 
Villars). 


24 SECOND LECTURE 


The two types of integro-differential equations, which we 
just enumerated, are completely different in their treatment. 
Volterra’s type resembles the partial differential equations (of 
the elliptic or sometimes parabolic genus in the examples hitherto 
given). The equation must be completed by accessory condi- 
tions which are nothing else than boundary conditions (cf. 
Lecture I). The methods given by Volterra run exactly parallel 
to those which are applied for Dirichlet’s problem (such as the 
formation of Green’s functions). 

In the second type described above, the accessory condi- 
tions are initial ones; and are to be treated in the manner, not 
of partial, but of ordinary differential equations—such methods 
as Picard’s successive approximations being of great use in that 
case. 


LECTURE II 


ANALYSIS SITUS IN CONNECTION WITH CORRESPONDENCES AND 
DIFFERENTIAL EQUATIONS 


it: 


We are going to speak of the réle of analysis situs in our 
modern mathematics. This theory is also called the geometry of 
situation. It is the study of connections between different parts 
of geometrical configurations which are not altered by any con- 
tinuous deformation. We suppose that we can let a system 
undergo any deformation whatever, however arbitrary it may be, 
only that it preserves continuity. For instance, a sphere and a 
cube are considered as one and the same thing from the point 
of view of the geometry of situation, because one can be trans- 
formed into the other without separating parts, or uniting parts 
which formerly were separated. The circle and the rectangle 
are identical from the same point of view. But the lateral 
surface of a cylinder and the surface of a rectangle are not 
identical, because, for the transformation of one into the other, 
we must make a cut along a generatrix. Also one is limited by 
two lines (the base circles) while the other is limited by one. 
The total surface of a cylinder is entirely closed; it is identical 
with the surface of a sphere. There is no difficulty in the 
transformation. 

If we consider the “anchor ring,” the case is different. 
This is a closed surface but it has a hole which is not found 
in the surface of the sphere, and the surface of the sphere can- 
not be transformed continuously in it. It would have to be 
transformed by several cuts, the first of them (Fig. 2) giving a 
broken ring, which for us is identical with the lateral surface of 


a cylinder. This may be cut into a rectangle and then trans- 
25 


26 THIRD LECTURE 


formed into a sphere. But the transformation of an anchor 
ring into a sphere cannot be done without cutting and piecing. 
The principles of analysis situs, for surfaces in ordinary space, 





Fia: 2. 


are well known and I do not intend to go over them at this mo- 
ment. We shall take them for granted. According to them, 


Fig. 3. 


a surface of two dimensions is defined from our present point of 
view by the number of boundaries and another number, namely 


ANALYSIS SITUS 27 


the genus. The genus is zero for the sphere and one for the 
anchor ring. For a pot with two “ears” (Fig. 3) we have the 
genus two. 

Analysis situs started with trifling problems, such as that 
treated by Euler of the bridges of Kénigsberg over the Pregel 
river. There are seven bridges; the problem is to go over all 
of them without passing twice over any one (Fig. 4). The great 





Fia. 4. 


Euler did not disdain to occupy himself with this and many 
other apparently childish problems. But what interests us in 
this one especially is that it involves the geometry of situation, 
in the sense in which we have used the term. For even if the 
islands in the river had other shapes and the bridges had the 
queerest forms, the reasoning would be exactly the same, pro- 
vided the numbers of islands and bridges should not change, and 
each bridge should join the same islands in both cases. 

We have here an example of an important theory which 
develops from a childish exercise. Some would think that it was 
a disadvantage to mathematics that we should occupy ourselves 
with such problems. The fact is, as we see, that they may, 
though exceptionally, lead to valuable results. 

That this notion of analysis situs was really an important one, 
appears first from the researches of Riemann. You know that 
Riemann was the fellow founder with Cauchy of the modern 
theory of analytic functions. These two schools applied their 


3 


28 THIRD LECTURE 


theories to the study of algebraic functions. Cauchy’s methods, 
in the hands of their author and of Puiseux, were capable of 
casting light on some important parts of the problem, but did 
not however completely elucidate it, and (in particular) Riemann 
alone could discover the fundamental notion of the genus of an 
algebraic curve. 

What were the elements of Riemann’s success and superiority 
over Cauchy? A remark must first be made which perhaps, 
strictly speaking, would not be within our subject, but which 
is nevertheless, as we shall see, most closely and necessarily 
connected with it. 

Let us consider the real domain. Suppose that we have to 
study the algebraic function y defined by 22+ y? = 1 (or any 
quadratic equation defining y as a function of.x corresponding to 
an ellipse). This function is real only for values of 2 which are 


Fra:'5, 


comprised between — 1 and + 1 (in the second case, for values 
between x and 2). Riemann considered the function in the seg- 
ment comprised between these values. He remarked that this 
is an incomplete view of the equation, for y is not well defined, 


ANALYSIS SITUS 29 


because it has two different values. But if we change our straight 
line into two slightly different straight lines, then we may admit 
that the superior segment corresponds to the + value of y, 
and the inferior one to the — value, the two segments being 
supposed to join each other at their common ends. To each 
point of the drawing, after that modification, one and only one 
system of values of a and y verifying the given equation will 
correspond. Besides, in that case, we obtain a figure which 
from the point of view of analysis situs, is identical with the 
ellipse represented by the given equation itself. 

But Riemann applied that same method in the complex 
domain, and was led to the celebrated kind of representing sur- 
faces which bear his name. 

This principle is a very general one. It must be applied, in 
any case, before using the geometry of situation. We must 
inquire whether the domain used is adequate to represent the 
states of variation to be studied. I shall give an instance which 
I think is due to Sophus Lie. It is concerned with the singular 
solution of differential equations of the first order. Given the 
differential equation 

f(x,y, y') = 9 (1) 
the question, as well known, is whether some solution exists which 
is not represented in the general integral. In that case such a 
solution must verify not only the original equation, but also 


ay’ a (2) 


Darboux showed that this was not sufficient, and that, in general, 
the system of equations (1) and (2) does not represent an actual 
solution, but that the curve which it defines is the locus of the 
cusps of the solutions of equation (1) (Fig. 5). We now shall 
see that this result, the analytical proof of which requires some 
complicated calculations, appears of itself by the above geo- 
metric considerations. 

Equation (1) defines y’ as a function of x and y, but this func- 


30 THIRD LECTURE 


tion has several determinations or branches. This state of things 
is not satisfactory from our point of view above. In order to 
avoid this, let us consider the surface f(x, y, 2) = 0 inspace. For 


each point of that surface, we have 


dy/dx = z (3) 







Saray, 


a ad 
i 


l 


Fia. 6. 


So that the problem becomes to trace on the surface, those curves 
which have dy/dx equal to z. Geometrically speaking, such 
curves must, in each point, be tangent to a certain direction, viz. 
the intersection of the tangent plane to the surface with a certain 
vertical plane (represented by (3)). The system (1) and (2) 


ANALYSIS SITUS 31 


represents the “horizontal boundary” of the surface. At each 
point m on it, the tangent plane is vertical (Fig. 6). What 
happens there? We see that in m, the two planes which define 
the tangent to our curve are vertical (the plane corresponding to 
(3) being so in any case). Therefore, this tangent itself is also 
vertical. This gives immediately the desired result; for it is 
well known that by projecting a space curve on a plane perpen- 
dicular to one of its tangents, we obtain a projection curve which 
has a cusp. The only exception would be when our two planes 
would coincide and- this indeed gives the supplementary con- 
dition for the existence of a singular solution. 

A difficult question in differential equations is thus recon- 
ducted to an elementary result of analytical geometry; and this 
is obtained by the mere fact of depicting correctly (in the sense 
of Riemann) y’ as a function of 2 and y.. Only when this ade- 
quate representation of the domain of variation is obtained, 
analysis situs is to be applied. 

Before seeing it in operation, let us notice that Cauchy had an 
opportunity of discovering its importance. This is a curious 
historical fact in his work; for it was one of his few errors. 
It was done in his youthful period, when dealing with the theorem 
of Euler on polyhedrons. This theorem connects the number of 
faces, summits and edges. It expresses that P+ V = E+ 2, 
where F is the number of faces, V is the number of vertices, and 
E the number of edges. Cauchy’s demonstration was false, 
and so is even the theorem itself. This theorem holds effectively 
(and this is the reason why Euler and Cauchy believed it to be 
true) for a very large category of polyhedra, among which every 
convex one occurs. But others had been overlooked, such as 
those which have the general shape of an anchor ring, and these 
do not verify the above relation. If Cauchy had perceived 
that error; if he had noticed that exception to Euler’s theorem, 
it may be presumed with some probability that he would not 
have left to Riemann the glory of founding a complete theory 
of algebraic functions. 


32 THIRD LECTURE 


Let me remind you of the difference between the method of 
Cauchy (and of Puiseux) and that of Riemann. If we consider 
the algebraic function defined by F(z, y) = 0, then y, in general, 
in the environs of a and yo, is a regular analytic function of a 
and is given by a Taylor’s series within a certain circle around a. 
Inside this circle, the principles of Cauchy and Weierstrass 
permit us to study the function. At critical points 21, where 
y is not a holomorphic function of 2, Puiseux studied this. 
He took X = (x — 2)'/”, p being properly chosen. Then y can 
be developed in powers of X instead of-in terms of 2 — a}. 
Everything seems at first to be settled then. But really we still 
ignore some fundamental properties. The reason of this is that 
we do not get the direct idea of the total domain, but only an 
indirect idea of it by a series of smaller regions. 

It is true that these smaller regions are such that, taken alto- 
gether, they cover the totality of the domain in question, and 
for that reason, they finally may enable us to master it com- 
pletely. But the error was to believe that this could be without 
a special study of the manner in which those partial regions 
are united. 

I should compare this (though the comparison is very in- 
complete) to the map of a large country, which is given by a 
series of partial leaves. We must take account, not only of 
each separate leaf, but of the “assembling table” showing their 
general disposition, so as to pass from the detail to the whole. 
The capital and unexpected fact, the discovery of which belongs 
to Riemann, is that such “assembling tables” are not at all 
like each other; that there are several quite different kinds of 
them: therefore, the synthesis of the details of the solution cannot 
be well understood without noticing these differences. 


2. 


It is now evident that the importance of these considerations 
is not limited to algebraic functions. They are connected with 
every synthesis of the above mentioned kind, that is to say, 


ANALYSIS SITUS 33 


theoretically speaking, with every employment of integral 
calculus. 

They constitute a sort of revenge of geometry on analysis. 
Since Descartes, we have been accustomed to replace each geo- 
metric relation by a corresponding relation between numbers, 
and this has created a sort of predominance of analysis. Many 
mathematicians fancy they escape that predominance and consider 
themselves as pure geometers in opposition to analysis; but most 
of them do so in a sense I cannot approve: they simply restrict 
themselves to treating exclusively by geometry questions which 
other geometers would treat, in general quite easily, by analytical 
means; they are of course, very frequently forced to choose 
their questions not according to their true scientific interest, 
but on account of the possibility of such a treatment without 
intervention of analysis. I am even obliged to add that some 
of them have dealt with problems totally lacking any interest 
whatever, this total lack of interest being the sole reason 
why such problems have been left aside by analysts. Of course, 
I not only admit geometrical treatment, but use it every time 
I find it possible, for, if applicable at all, it gives us, in general, a 
much better view of the subject than an analytical one. But 
very important problems may be inaccessible to it. We must 
use all means at our disposal and choose, not this or that one 
a priori, but the one best adapted to our question. 

But here geometry has over analysis a more certain ad- 
vantage. I consider that analysis could not, or could only 
with great difficulty, and probably after a long series of sterile 
efforts, have replaced the geometrical views we have just alluded 
to for resolving the corresponding part of the problem. I mean 
that passage from the solution in small regions to the solution 
over the whole domain." 

1 Logically speaking, even the results of analysis situs can be rigorously 
stated in numerical language; but such statements have been made only 


after the results have been found, and some parts of this analytic treatment 
are of extreme difficulty (such as Jordan’s theorem), 


34 THIRD LECTURE 


Let us, for instance, admit that that domain is a two-dimen- 
sional one. Then according to analytical methods, we ought to 
individualize any point of it by giving the values of two param- 
eters, x and y. But the representation of a geometrical 
problem by means of functions of 2 and y often makes us lose 
some element of the problem: functions in a domain in two 
dimensions may be something else than the functions of 2 
and y. The simultaneous variation of « and y represents a 
plane. Now a plane has not the same general shape as a sphere 
or anchor ring, and those differences are lost in Descartes’s 
method. We can have, for instance, as many examples of this 
difference in rational dynamics as we please. One knows that 
when a dynamical problem has two degrees of freedom the corre- 
sponding differential equations, i. e. the equations of Lagrange, 
are defined, the parameters which define the position of the 
system being designated by 2 and y, if one gives the expres- 
sion 27 = E(a, y)a” + 2F(a, y)x'y' + G(x, y)y” for the vis viva 
and the expression U = ¢(z, y) for the force function. 'There- 
fore, if two problems of dynamics correspond to the same ex- 
pression of 7' and the same expression of U, their studies ought 
to be exactly identical and reducible to each other. That mat- 
ters may really be quite different is to be immediately seen 
by the following example: 

(1) Consider the material particle acted on by no forces. 
The trajectories will be straight lines. (2) Let us have a vertical 
standard. The arms AA’ and BB’ are solidly attached and 
A and B are fixed (Fig. 7). The only motion of the system is 
a rotation about AB. A’B’ is a second axis about which a rigid 
body homogeneous and of revolution can rotate. The system 
has two degrees of freedom. We have to study the motion of the 
system. There will be no force function. Only rotations are 
possible (two independent ones around AB and one around A’B’). 

Analytically, the two problems are one and the same, for in 
both cases, U = 0 and the coefficients L, F, Gin 2T are constants 
(which can always, by a linear transformation on 2, y, be reduced 


ANALYSIS SITUS 35 


to H=G=1, F=0). Nevertheless, there is evidently no 
comparison between the motions in case (1) and case (2), so 
that to a certain extent, we are deceived by analytic methods. 
The assemblage of all possible positions of system (2) can be 
represented not on a plane, but on the surface of an anchor ring. 





Fig. 7. 


We know since the researches of Poincaré that the study of 
trajectories represented by differential equations must be founded 
on analysis situs. For instance, f(z, y, y’) = 0 is geometrically 
represented by a certain surface, and on this surface defines a 
geometrical correspondence as follows: for each point of the 
surface it defines a certain direction (with its sense) in the 
tangent plane. We have then to draw at each point of the sur- 
face a curve which is tangent to the direction thus defined. 


36 THIRD LECTURE 


Poincaré showed that such a problem cannot be handled unless 
we know what the genus of the surface is. This already appears 
in a simple preliminary question which arises in that study. We 
have said that we have a certain direction at each point of our 
surface. Can we in general do this without exception? In 
general we cannot. In each point, in general, we shall have a 
certain tangent direction defined, but there will be certain 
singular points in the correspondence. The only case in which 
the correspondence can be complete is when the surface is of 
genus one. For instance, there must be singular points for the 
genus zero. In that case, Poincaré stated that every trajectory 
is either a closed one, or finishes in a singular point, or is asymp- 
totic to a closed curve. For genus one, singular points may be 
absent, but the shapes of curves verifying the equation may 
yet be much more complicated. 

Differential equations of higher order will also of course (and 
did indeed in some parts of Poincaré’s work) require the inter- 
vention of analysis situs. But the difficulty will be much greater, 
as in hyper-spaces this theory becomes as complicated as it was 
simple in Riemann’s hands when applied to ordinary surfaces. 
These higher chapters of analysis situs begin, however, to be well 
known, and though they could not hitherto be applied to differ- 
ential equations, their réle is already clear, owing to the works 
of Picard and Poincaré, in the natural generalization of Riemann’s 
original theory. I mean the difficult theory of algebraic surfaces 
and algebraic functions of two or more independent variables. 

In the line of partial differential equations, We must point out 
a very remarkable analogous example due to Volterra and con- 
cerning the problem of elasticity. Generally speaking, if the 
external forces and also the peripheric efforts acting on a homo- 
geneous solid body are zero, so will be the stress at every point 
of its substance. More precisely in such a body of simply con- 
nected shape, stress could only appear under those conditions if 
singular points would exist where they would cease to obey the 
general laws known for their distribution. But the contrary can 


ANALYSIS SITUS a6 


take place if the body has an annular form, and in fact Volterra 
practically constructed such annular bodies in which stress exists 
and can be experimentally perceived, without any external action 
and without any singular point. 


3. 


But examples of a much more elementary character, belonging 
to the very beginning of the differential calculus, can be given. 
Let us consider a point-to-point correspondence, defined by such 
equations as 

X = f(z, Y)s Y= g(x, y)- 
When does that system of equations admit one and only one 
solution in x, y if X, Y are supposed to be given? 

It is classical that this, above all, depends on the functional 
determinant 

of of 
DEX Y) 7 Ox Oy 
D(x, y) Og Og 

Ox dy 





Suppose that this is not zero in a certain point a, yo. We are 
taught that in the neighborhood of (Xo, Yo) the system will have 
one and only one solution. The tempting conclusion is to 
suppose that if everywhere this determinant is not zero, then 
everywhere we will have a one-to-one correspondence. This is 
not true, and indeed errors have been committed on that subject. 
Even in the simplest case, in which the representation of the 
whole plane of XY on the whole plane of xy is considered, a sup- 
plementary condition at infinity must be added in order to 
ascertain that the transformation is one-to-one. 

But now let us replace our planes by two spheres, a corre- 
spondence being considered between a point (x, y, 2) of the surface 
of the first sphere, and a point (XY, Y, Z) of the surface of the 
second. In this case we find that if a condition analogous to 
that above holds at every point of the first surface it will actually 
insure a regular one-to-one correspondence. 


38 THIRD LECTURE 


But if we replace our spheres by two anchor rings, the results 
will again be completely and utterly changed. Several points 
on the surface of one anchor ring may correspond to one and the 
same point on the surface of a second one, although in the 
neighborhood of each point everything seems to take place just 
as in a one-to-one correspondence. ‘To see this, one has only 
to note that a point on the torus depends on two angles, 9, ¢. 
If we call 0’, ¢’ the two similar angles for the second surface, 
we have only to define the correspondence by 0’ = 70, ¢’ = q¢, 
p and q being two arbitrary integers.! 

A curious fact is that the same thing occurs with respect to 
two circles. It is evident that if two points respectively move 
on the two circumferences with uniform speed, one turning 
exactly p times (p being an integer) while the other turns once, 
each position of the former will correspond to p distinct positions 
of the latter, although the ratio of speeds never changes signs, 
nor even becomes zero or infinite. 

Nothing of the kind could, as we saw, occur on the surfaces 
of our two spheres (nor of two hyperspheres in n-dimensional 
space, if n > 2), so that, in that respect, the case of two dimen- 
sions proves more complicated than that of three or more 
dimensional spaces. 

These peculiar distinctions are closely connected with the fun- 
damental distinctions of analysis situs. They are due to the fact 
that there are many ways essentially distinct from each other, of 

1]t is interesting to add that as far as ordinary (closed) surfaces are con- 
cerned, the genus 1 is the only one for which such a paradoxical circumstance 
can occur, in the sense that, if each point of a closed surface 2, of genus g > 1, 
corresponds to one (and only one) point of a second closed surface D’ of the 
same genus, and if, in the neighborhood of each point, the relation thus defined 
takes the character of a one-to-one regular correspondence, it is such on the 
whole surfaces. 

This is easily seen in noting that, more generally, if we place ourselves 
under the same conditions except that we do not suppose the two genera, 
g, g’ to be equal, and if h be the number of points of 2 corresponding to same 
point on = this number h (which must be the same everywhere, on account of 


the absence of singular points) is connected with g, g’ by the equation 
g —1=h(g’ — 1): a fact which results from the generalized Euler’s theorem. 


ANALYSIS SITUS 39 


passing from one point to another of a circumference (according 
to the number of revolutions performed around the curve) whilst 
any line joining two points of the surface of a sphere can be 
changed into any other one by continuous deformation. 

This question of correspondences and Euler’s theorem on 
polyhedra would give us the most simple and elementary in- 
stances in which the results are profoundly modified by con- 
siderations of analysis situs, if another one did not exist which 
concerns the principles of geometry themselves. I mean the 
Klein-Clifford conception of space. But since this conception 
has been fully and definitively developed in Klein’s Evanston 
Colloquium, there is no use insisting on it. We want only to 
remember that this question bears to a high degree the general 
character of those which were spoken of in the present lecture. 
Klein-Clifford’s space and Euclid’s ordinary space are not only 
approximately, but fully and rigorously identical as long as 
the figures dealt with do not exceed certain dimensions. Nothing 
therefore can distinguish them from each other in their infini- 
tesimal properties. Yet they prove quite different if sufficiently 
great distances are considered. 

This example, as you see, exactly like the previous ones, 
teaches us that some fundamental features of mathematical 
solutions may remain hidden as long as we confine ourselves 
to the details; so that in order to discover them we must neces- 
sarily turn our attention towards the mode of synthesis of those 
details which introduce the point of view of analysis situs. 


LECTURE IV 


ELEMENTARY SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS 
AND GREEN’S FUNCTIONS 


1. Elementary Solutions 


The expressions we are going to speak of are a necessary base 
of the treatment of every linear partial differential equation, 
such as those which arise in physical problems. The simplest 
of them is the quantity employed in all theories of the classical 
equation of Laplace: V’u = 0; namely the elementary Newtonian 
potential 1/r, where 


r= V(x —a)?+ (y— b+ (z— ©)? 


and (a, 5, ¢) is a fixed point. 

The potential was really introduced first and gave rise to the 
study of the equation. All known theories of this equation 
rest on this foundation. The analogous equation for the plane is 





“2 uOt 
dx? ay? ° 
Here we must consider the logarithmic potential, log 1/r, where 
r= V(e—a?+ (y— b). By this we see that if we wish 
to treat any other equation of the aforesaid type, we must try 
to construct again a similar solution which possesses the same 
properties as 1/r possesses in the case of the equation of Laplace. 
How is such a solution to be found? 'To understand it, we must 
examine certain properties of 1/r. First let us note that that 
quantity 1/r is a function of the codrdinates of two points 
(x, y, 2) and (a, b, c) [the corresponding element log 1/r in the 
plane being similarly a function of (a, y; a, b)]. If considered 
as a function of 2, y, z, alone (a, b, c, being supposed to be con- 
stant) in the real domain, 1/r is singular for r = 0; and r = 0 
40 





ELEMENTARY SOLUTIONS Al 


only when «=a, y= 6b and z =e simultaneously. But for 
complex points, 1/r is singular when the line that joins (a, y, 2) 
and (a, b, ¢) is part of the isotropic cone of summit (a, b, c). 

This isotropic cone is not introduced by chance, and not any 
surface could be such a surface of singularity. It is what we 
shall call the characteristic cone of the equation. We already 
met with the notion of characteristics in our first lecture, and 
saw that it is nothing else than the analytic translation of 
the physical expression “waves.”’ I must nevertheless come 
back to it this time in order to remind you that the word 
“waves” has two different senses. The most obvious one is the 
following: Let a perturbation be produced anywhere, like sound; 
it is not immediately perceived at every other point. There are 
then points in space which the action has not reached in any 
given time. Therefore the wave, in that sense a surface, 
separates the medium into two portions (regions): the part 
which is at rest, and the other which is in motion due to the 
initial vibration. These two portions of space are contiguous. 
It was only in 1887 that Hugoniot, a French mathematician, 
who died prematurely, showed what the surface of the wave can 
be; and even his work was not well known until Duhem pointed 
out its importance in his work on mathematical physics. 

A second way of considering the wave is more in use among 
physicists. We have not in the first definition implied vibrations. 
If we now suppose that we have to deal with sinusoidal vibra- 
tions of the classical form, the motion is general and embraces 
all the space occupied by the air. Tracing the locus of all 
points of space in which the phase of the vibration is the 
same, we determine a certain wave surface (or surfaces). 

It is clear that these two senses of the word “waves” are 
utterly different. In the first case, we have space divided into 
two regions where different things take place, which is not so 
in the second case. Certainly, physically speaking, we feel a 
certain analogy between them. But for the analyst, there seems 
to be a gap between the two points of view. 


42 FOURTH LECTURE 


The gap is filled by atheorem of Delassus. Let us consider any 
linear partial differential equation of the second order, and sup- 
pose that w is a solution which would be singular along all points 
of a certain surface, 7(a, y, 2) = 0. By making some very simple 
hypotheses as to the nature of the singularity, Delassus found 
that this surface must be a characteristic as defined in our first 
lecture; that is, it must verify, if the given equation is Y’u = 0, 
the (non-linear) partial differential equation of the first order 


On \2 On \?2 Or \2 
(32) + (3) + (E)=° 


obtained by substituting for the partial derivatives of the second 
order of the unknown function w in the given equation, the 
corresponding squares or products of derivatives of the first 
order of z (the other terms of the given equation being considered 
as cancelled). This is the characteristic equation corresponding 
to our problem. It is the same as the one found by Hugoniot 
in studying the problem from the first point of view. This third 
definition will show us the connection between the first two. In 
the first case, the wave corresponds to discontinuity, for the 
speeds and accelerations change suddenly at the wave surface: 
such a discontinuity is evidently a kind of singularity. In the 
vibratory motion the general equation contains the factor 
sin wr since u = F sin uz, where F is the parameter corresponding 
to the frequency, and 7 is a function of 2, y, z. This form of u 
seems to show no singularity, for the sine is a holomorphic function 
It is nevertheless what one may call “practically singular.” If 
we suppose that the absolute magnitude of u is large, the function 
varies very rapidly from + 1 to — 1, it has derivatives which 
contain yw in factor, and these derivatives are therefore very 
large. It has a resemblance to discontinuous function because 
of the large slope. So that, in what may be called “approxima- 
tive” analysis, it must be considered as analogous to certain 
discontinuous functions. From that point of view the three 
notions of waves are closely connected. 


ELEMENTARY SOLUTIONS 43 


This view of Delassus is the one which will interest us now 
because in the case of the elementary solution 1/r the char- 
acteristic cone is a surface of singularity. We see now in what 
direction we may look for the solution of the problem. We 
have to find what will be the characteristic cone or surface 
corresponding to it. Then we must construct a solution having 
this as a singularity. The first question is answered by the 
general theory of partial differential equations of the first order. 
We must have a conic point at (a, b,c). In general the char- 
acteristic cone is replaced by a characteristic conoid which has 
curvilinear generatrices which correspond to the physical “rays.” 
Secondly, we must build a solution which will have this for a 
surface of singularity. The first work of general character in this 
direction was that of Picard in 1891. He considered the case 
of two variables and treated more especially the equation 





: un 60 
a) cae 
Not every equation of the general type 
ru 07u ru du du 
—= 9 == —. = 
Anat crea Os pat 2D Peis Toe ae Fu = 0 


can be reduced to that form. But in the elliptic case (B? — AC 
< 0) it can, by a proper change of independent variables, be 
reduced to the form 


au , du Ou Ou 
(1) yee ap. an ena a de 


(in which the characteristic lines are the isotropic lines of the 
plane). Sommerfeld and Hedrick treated this more general 
form and showed for equation (1), as Picard had done for the 
equation (1’), that there exists an elementary solution, possessing 
all the essential properties of log 1/r. It is 


P log 1/r+Q 
P and Q being regular functions of x and y. P has the value 1, 
4 


44 FOURTH LECTURE 


«=a, y=b. In the hyperbolic case (real. characteristics), 
the form to which the equation can be reduced. is Laplace’s form 


Ou 


Ou Ou 
2) dxdy fe Ou ae oy eats 4h 





if the change of variables is real; and the corresponding ele- 
mentary solution is of the type 


P log Vx — a)(y— 5) + Q 

P and Q having the same significations as above (P is nothing 
else than the function which plays the chief réle in Riemann’s 
method for equation (2)). Of course, if imaginary changes were 
admitted (which is possible only if the coefficients are supposed 
to be analytic) elliptic equations, as well as hyperbolic ones, 
could be reduced to the type (2) or as well, (1). The only 
case in which that reduction is not at all possible, is when 
B? — AC = 0, the parabolic case. This is a much more difficult 
case. It has been treated only recently. There is a new type 
of elementary solution which was given in 1911 by Hadamard in 
the Comptes Rendus, and for the equation of heat with more than 
two variables by Georey that same year (in the same periodical). 

Even if we leave the parabolic case aside, the question has a 
new difficulty arising because it is not possible to simplify by 
changing variables as before when there are more than two of 
them, so that we must then treat the general case. The problem 
was, however, first treated in the case of 





Ou Ou Ou 
2 : =~ 
Mae amb ary ester ae 840 


But not every partial differential equation of the second order in 
three variables can be reduced to this form. It is important 
nevertheless. Holmgren obtained a solution in form analogous 
to 1/r, namely P/r, where P = 1 forr = 0. 

If we wish to treat the general case where the coefficients are 
quite arbitrary, we must try first to form the surface of singu- 
larity which is the characteristic conoid. Suppose first that we 


ELEMENTARY SOLUTIONS 45 


have any regular characteristic surface of our equation and 
suppose that by a change of variables, x = 0 is the surface. 
Let us write w= x?F (a2, y, z). One can show that, giving p 
any positive value, solutions of this form can be found, F being 
regular. Such is not the case when p is a negative integer; and 
this gives us again an interesting illustration of the consider- 
ations explained in our first lecture in connection with Schoenflies’ 
theorem. Let p be a negative integer and suppose that there is 
a solution. Then we have also other values of wu of the form 


ne 


e 


a Nghe Yy; Z) 


(We can form an infinity of these solutions because the differential 
equation possesses an infinity of regular solutions.) But those 
values of wu can be written 


F+ 2? Fy 


eP 


So that, if our question is possible, it has an infinity of solutions. 
By the same reasoning as in the first lecture, we must not wonder 
at its being in general not possible. There is again this balancing 
between infinity of solutions and their existence. 

But we have supposed our characteristic surface to be a 
regular one. If we deal with our characteristic conoid, which 
has (a, b, c) for a conic point, things behave differently; p cannot 
have an arbitrary value. If the number of independent vari- 
ables is n, we must have 


= 9 aia a 
Tae ee Cee 
Deere mem heey OF - ( 9 +1), - (“3 4+2),- 


= 





The first of these values is, however, the only essential one, 
because, if we have formed the (unique) solution corresponding 
to p= — (n— 2)2, which depends on 2, y, 2, a, b, c, we can 
deduce all others from it: we need merely to differentiate with 
respect to a, b, c. 

If n is even, those values of p become negative integers and 


46 FOURTH LECTURE 


therefore, on account of what we just said, there is, in general, 
no solution of the above form ; 


w= A+ 


We have to replace this by 
hs 
U= i +. Ie, log iP 


in which T would again be equal to r’, r meaning a distance in 
n-dimensional space, if the higher terms (of the second order) 
of the given equation are of the form V’u. However, if these 
terms are arbitrary, I should be replaced by the first member 
of the equation of the characteristic conoid of summit (a, b,c). 

The functions P, Q, P: can easily be developed in convergent 
Taylor’s series if the coefficients of the equation are analytic. 
If not, they still exist but are much more difficult to find. The 
first result of Picard, concerning the special equation (1’), was 
however, obtained (by successive approximations) without any 
assumption on the analyticity of c: Later, E. E. Levi solved the 
problem in the same sense for the general elliptic equation. 

The principle of these methods of Picard and Levi in reality 
is the same. Both may be considered as peculiar cases of one 
indicated by Hilbert and consisting in the introduction of the 
first approximation, which presents a singularity of the required 
form, but does not need to verify the given equation. The 
investigation of the necessary complementary term leads 
again to an integral equation. I must add that, for equa- 
tions of a higher order, the extension of this seems to offer 
difficulties of an entirely new kind, owing to the fact that the 
characteristic conoid generally admits other singularities than its 
summit (viz. cuspidal lines). For the very special case in which 
there are no other terms than those of the highest order, the 
coefficients of those terms being constant, it has however been 
reduced to Abelian integrals by a beautiful analysis of Fredholm’s, 


ELEMENTARY SOLUTIONS 47 


2. Green’s Functions 


Elementary solutions are a necessary instrument for the 
treatment of the partial differential equations of mathematical 
physics. They are not always sufficient. They are sufficient 
for the simplest of the problems alluded to in our first lecture, 
namely Cauchy’s problem. But we know that for the ellip- 
tic case, this latter is not to be considered, and we have to 
face others, such as Dirichlet’s problem. For Dirichlet’s prob- 
lem (i. e. to find uw taking given values all over the surface of 
the volume S, and satisfying V’u = 0), 1/r is not a sufficient 
function. We must introduce a new function of the form 1/r + h 
where h is a regular function; and h must be such that 1/r + h 
must be zero at every point of the boundary surface. This is 
called Green’s Function. It is the potential produced on the 
surface S by a quantity of electricity placed at (a, b, c) interior 
to the surface, this surface being hollow, conducting, and main- 
tained at the potential zero. This is its physical interpretation. 

For any other linear partial differential equation of the elliptic 
type, one has to consider such Green’s functions in which the 
term 1/r is to be replaced by the elementary solution (so that; 
at any rate, the formation of this latter is presupposed), fi still 
being a regular function (at least as long as (a, b, c) remains fixed 
and interior to S). 

Similar sorts of Green’s functions are also known for bigher 
differential equations, e. g. for the problem of an elastic plate 
rigidly fastened at its outline, the differential equation being 
then V?v2u = 0 (in two variables x and y only) and the rdle of 
elementary solution being played by 7? log r. 

Like 1/r and like the elementary solution itself, any Green’s 
function depends on the codrdinates of two points, A(x, y, 2) 
and B(a, b, c). But the chief interest in the study of those 
Green’s functions, the important difference between them and 
the above mentioned fundamental solutions, corresponds to a 
similar difference between Cauchy’s and Dirichlet’s problems, 
such as defined in our first lecture. To understand this, let us 


48 FOURTH LECTURE 


remember that each of those two problems depends on three 
kinds of elements: 

1. A given differential equation; 

2. A given surface (or hyper-surface in higher spaces) S; 

3. A certain distribution of given quantities at the different 
points of S. 

Each of those elements has of course its influence on the 
solution but not to the same degree. The influence of the form 
of the equation cannot but be a profound one. On the contrary, 
the influence of the quantities mentioned in 3 is comparatively 
superficial, in the sense that the calculations can be carried pretty 
far before introducing them. In other terms, if we compare this 
to a system of ordinary linear algebraic equations, the réle of 
the first element may be compared to that of the coefficients of 
the unknowns (by the help of which such complicated expres- 
sions as the determinant and its minor determinants must be 
formed) while the réle of the third element resembles that of the 
second members which have only to be multiplied respectively 
by the minor determinants before being substituted in the 
numerator. 

But as to the réle of our second element, the shape of our 
surface S, the answers are quite different according to cases. 

If we deal with Cauchy’s problem, that shape plays just as 
superficial a réle as the third element. For instance, in Rie- 
mann’s method for Cauchy’s problem concerning equation (2), 
every element of the solution can be calculated without knowing 
the shape of S (which in that case is replaced by a curve, the 
problem being two-dimensional) till the moment when they have 
to be substituted in a certain curvilinear integral which is to be 
taken along S. 

But matters are completely different in that respect in the 
case of Dirichlet’s problem. While one can practically say that 
there is only one Cauchy’s problem for each equation, there is, 
for the same and unique equation V’u = 0, one Dirichlet’s 
problem for the sphere, one for the ellipsoid, one for the paral- 


ELEMENTARY SOLUTIONS 49 


lelepipedon; and these different problems present very unequal 
difficulties. 

It is clear that the same differences will appear in the mode 
of treatment corresponding to the two problems. The elemen- 
tary solution depends on nothing else than the. given equation 
and the codrdinates 2, y, 2, a, b, c, of the two points A, B. 

The Green’s function on the contrary depends, not only on 
this equation and these codrdinates, but also on the form of 
the boundary S. 

The interesting question arising therefrom is to find how the 
properties of Green’s functions are modified by the change of 
the shape of the surface. Let us replace S by S’, defined by its 
normal distance 6n (which may be variable from one point of S 
to another). Take two given points A and B within S. Then 
there is a certain form of Green’s function g% for the surface S, 
and if we change from S to S’, g% changes. The change is 


(3) 693 = { ee snd 
dg”A., ‘ 
dn the rate of change of g, relative to the change of n. 

Here dndS is an element of volume comprised between the 
surfaces 8S, 8’. Similar formulas hold for Green’s functions for a 
planearea. They are like those given by the calculus of variations 
of integrals, though its methods are not directly applicable. 

A curious consequence is that from all the Green functions 
for all the elliptic partial differential equations, we can deduce 
by proper differentiations expressions verifying one and the 
same integro-differential equation, namely 


So3 = So7.p25ndS 


The fact that in the second member of the equation (3), the 
coefficient of dndS is quadratic and symmetric with respect to 








1 All these observations quite similarly hold for the “mixed problems” 
alluded to in our first lecture, and for the expressions introduced in their 
treatment corresponding to Green’s functions. 


50 FOURTH LECTURE 

expressions depending on the points A and B respectively, is also 
an important one. Useful inequalities, which could not easily 
be obtained otherwise, can be deduced therefrom. 

Besides that study of the variation of the numerical values 
of Green’s functions, the influence of the shape of S can be 
studied from another point of view, I mean its influence on their 
analytical properties, and this has been the occasion for important 
recent results. The complementary term h in a Green’s function 
remains regular as long as one of the points remains fixed and 
interior to the considered domain; but it offers a peculiar 
singularity when the two points A, B simultaneously approach 
the same point P of the boundary; and that singularity looks 
at first like a very difficult one. Its study is nevertheless 
simplified by the fact that it only depends on the shape of S 
in the immediate neighborhood of P. 





Fia. 8. 

In the case of the plane, for instance, if two closed contours 
S, S’, limiting two different areas have a certain are MN in 
common! (Fig. 8), if P is a point of this arc, and if G, G’ be the 
two Green’s functions corresponding respectively to those con- 
tours, the difference G — G’ will be a completely regular function 
(admitting a development in a convergent Taylor’s series) when 
A and B are both very near to P. 


1The two contours are understood to be one and the same side of that 
are MN. 


ELEMENTARY SOLUTIONS BL 


We have now to inquire what the singularity of G, for instance, 
will be. After having received a first partial answer in interest- 
ing papers by several Italian geometers, this question has been 
completely solved by E. E. Levi for a function analogous to the 
ordinary Green’s function, and more recently by P. Levy for 
this latter itself. 

The answer thus obtained is remarkably simple in the case 
of twodimensions. P. Levy also works out the three-dimensional 
problem, but there the results are much more complicated. 

As to Green’s function as a whole (and not only the singular 
part of it) it must be well understood that its value for any two 
given points of the area or even such elements as its normal 
derivative in one point of the contour, profoundly depends on 
the form of every part of this latter, however distant from the 
point or points in question. 

By paying attention to this fact, we must expect, on account 
of what was seen in the preceding lecture, that considerations of 
analysis situs will be important in that question. At first this 
does not seem to be the case, and the most important methods 
for the resolution of Dirichlet’s problem are common to areas 
of any genus (although with some modifications of detail, as 
will be seen for Fredholm’s method in Kellogg’s Dissertation). 
But other views of the problem will show that the influence of 
analysis situs does exist here and is perhaps even more astonish- 
ingly profound than in any of the questions examined in our last 
lecture. 

If we consider again Dirichlet’s problem for an area in the 
plane, we shall see that the analytical properties of the corre- 
sponding Green’s function are very different if that area has one 
or several boundaries. 

Let us take the first case. In this case, the plane area can 
be represented conformally on a circle of unit radius with the 
origin as center. It is easily seen that, in such a conformal 
representation, Green’s function keeps its values, and this brings 
to light a remarkable consequence concerning the six Green’s 


52 FOURTH LECTURE 


functions generated by four points taken two by two. The 
six quantities have a relation between them and give rise to a 
peculiar sort of geometry, which not only resembles the ordinary 
non-Euclidean geometry, but can be reduced to it by a simple 
transformation. 

In an area with two boundaries (annular area) matters are 
quite different. Schottky has shown that if we take two such 
areas, S, 8S’, having each two boundaries, they are not in general 
conformally representable on one another. Each one of them 
will be represented on the area between two concentric circles. 
But the ratio of the radii of these circles must, in each case, be 
chosen properly, and, therefore, will not, in general, be the same 
for = and for 2’. 

In this last case, the relation between the six Green functions 
will not hold, and the properties of our Green’s functions will be 
far less simple. They will become still more complicated for 
more than two boundaries. We again have here an important 
instance of the réle played by analysis situs in analytical prop- 
erties, and as we have stated that Green’s functions are related 
to all the chief topics treated in our preceding lectures, this is 
perhaps the best conclusion to be given to the ensemble of them. 








Columbia University Press 


Columbia University in the City of New York 


Lemcke & Buechner, Agents 


30-32 West 27th Street NEW YORK 





Publications of the 
Ernest Kempton Adams Fund for Physical Research 


These publications are distributed under the Adams Fund to many libraries 
and toa limited number of individuals, but may also be bought at cost from the 
Columbia University Press. 


Number One. Fieldsof Force. By VinuetmM Friman Koren Bserxnes, Professor of Physics 
in the University of Stockholm. A course of lectures delivered at Columbia Univer- 
sity, 1905-6. 


Hydrodynamic fields. Electromagnetic fields. Analogies between the two. Supplementary lecture on 
application of hydrodynamics to meteorology. 160 pp. $1.00. 


Number Two. The Theory of Electrons and its Application to the Phenomena of Light and 
Radiant Heat. By H. A. Lorentz, Professor of Physics in the University of Leyden. 
A course of lectures delivered at Columbia University, 1906-7. With added notes. 
332 pp. Edition exhausted. Published in another edition by Teubner. 

Number Three. Eight Lectures on Theoretical Physics. By Max Puancx, Professor of 
Theoretical Physics in the University of Berlin. A course of lectures delivered at 
Columbia University in 1909, translated by A. P. Wiuus, Professor of Mathematical 
Physics in Columbia University. 


Introduction: Reversibility and irreversibility. Thermodynamic equilibrium in dilute solutions. 
Atomistic theory of matter. Equation of state of a monatomic gas. Radiation, electrodynamic theory. 
Statisticaltheory. Principle ofleast work. Principle of Relativity. 130 pp. $1.00. 


Number Four. Graphical Methods. By C. Runan, Professor of Applied Mathematics in the 
University of Gottingen. A course of lectures delivered at Columbia University, 
1909-10. 


Graphical calculation. The graphical representation of functions of one or more independent variables. 
The graphical methods of the differential and integral ca'culus. 148 pp. $1.50. 


Number Five. Four Lectures on Mathematics. By J. HapamMarp, Member of the Institute, 
Professor in the Collége de France and in the Ecole Polytechnique. A course of lectures 
delivered at Columbia University in 1911. 

Linear partial differential equations and boundary conditions. Contemporary researchesin differen- 


tial and integral equations. Analysis situs. Elementary solutions of partial differential equations 
and Green’s functions. 53 pp. 


Nuniber Six. Researches in Physical Optics, Part I, with especial reference to the radiation 
ofelectrons. By R. W. Woop, Adams Research Fellow, 1913, Professor of Experimental 
Physics in the Johns Hopkins University. 134pp. With10plates. Edition exhausted. 

Number Seven. Neuere Probleme der theoretischen Physik. By W. Wrmn, Professor of 
Physics in the University of Wiirzburg. A course of six lectures delivered at Columbia 
University in 1913. 


Introduction: Derivation of the radiation equation. Specific heat theory of Debye. Newer radiation 
theory of Planck. Theory of electric conduction in metals, electron theory for metals. The Einstein 
fluctuations. Theory of Réntgenrays. Method of determining wave length. Photo-electric effect and 
emission of light by canal ray particles. 76 pp. 



















Pik 





' 
a 


‘ie Toy 7 

eee ay 

s at iy TT 
i 


yy ’ a is 
» er i 
Maoh ae ; 
¥ : 





cm ~~? 2 7 a 
. = Joanie 
Meg 2 7 
a _* ’ 
+ 
am 


nee 





UNIVERSITY OF ILLINOIS-URBANA 


Q,515.353H11F co02 
FOUR LECTURES ON MATHEMATICS NEW YORK 


gu 





